# American Institute of Mathematical Sciences

May  2015, 20(3): 833-852. doi: 10.3934/dcdsb.2015.20.833

## Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate

 1 Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov, 61022, Ukraine 2 Institut für Mathematik, Institut für Stochastik, Ernst Abbe Platz 2, 07737, Jena, Germany

Received  September 2013 Revised  February 2014 Published  January 2015

We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
Citation: Igor Chueshov, Björn Schmalfuß. Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 833-852. doi: 10.3934/dcdsb.2015.20.833
##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar [2] G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163. doi: 10.1007/s00245-006-0884-z. Google Scholar [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. Google Scholar [4] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55. doi: 10.1090/conm/440/08476. Google Scholar [5] H. Bauer, Probability Theory,, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, (1991). doi: 10.1515/9783110814668. Google Scholar [6] P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten,, Thesis, (1988). Google Scholar [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). doi: 10.1007/BFb0087685. Google Scholar [8] A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar [10] I. Chueshov, Monotone Random Systems Theory and Applications,, Lecture Notes in Mathematics, (1779). doi: 10.1007/b83277. Google Scholar [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Methods Appl. Sci., 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar [12] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. Google Scholar [13] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. Google Scholar [14] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Well-posedness and long-time dynamics, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [15] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355. doi: 10.1023/A:1016684108862. Google Scholar [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Commun. Pure Appl. Anal., 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in System Modeling and Optimization (25th IFIP TC7 Conference, (2013), 328. doi: 10.1007/978-3-642-36062-6_33. Google Scholar [19] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [20] G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Ann., 331 (2005), 41. doi: 10.1007/s00208-004-0573-7. Google Scholar [21] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. Google Scholar [22] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Appl. Anal., 88 (2009), 1053. doi: 10.1080/00036810903114841. Google Scholar [23] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). Google Scholar [24] N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1997), 459. Google Scholar [25] J. E. Lagnese, Modelling and stabilization of nonlinear plates,, in Estimation and Control of Distributed Parameter Systems (Vorau, (1990), 247. Google Scholar [26] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1988). Google Scholar [27] J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl. (9), 85 (2006), 269. doi: 10.1016/j.matpur.2005.08.001. Google Scholar [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [29] T. J. Pedley, The Fluid Mechanics of Large Blood Vessels,, Cambridge University Press, (1980). doi: 10.1017/CBO9780511896996. Google Scholar [30] K. Petersen, Ergodic Theory,, Cambridge Studies in Advanced Mathematics, (1983). doi: 10.1017/CBO9780511608728. Google Scholar [31] B. Schmalfuß, The random attractor of the stochastic Lorenz system,, Z. Angew. Math. Phys., 48 (1997), 951. doi: 10.1007/s000330050074. Google Scholar [32] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar [33] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, [2013 reprint of the 2001 original] [MR1928881], (2001). Google Scholar [34] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984). Google Scholar [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995). Google Scholar [36] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar [2] G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163. doi: 10.1007/s00245-006-0884-z. Google Scholar [3] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. Google Scholar [4] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55. doi: 10.1090/conm/440/08476. Google Scholar [5] H. Bauer, Probability Theory,, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, (1991). doi: 10.1515/9783110814668. Google Scholar [6] P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten,, Thesis, (1988). Google Scholar [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977). doi: 10.1007/BFb0087685. Google Scholar [8] A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar [10] I. Chueshov, Monotone Random Systems Theory and Applications,, Lecture Notes in Mathematics, (1779). doi: 10.1007/b83277. Google Scholar [11] I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Methods Appl. Sci., 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar [12] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. Google Scholar [13] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008). doi: 10.1090/memo/0912. Google Scholar [14] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Well-posedness and long-time dynamics, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar [15] I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355. doi: 10.1023/A:1016684108862. Google Scholar [16] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Commun. Pure Appl. Anal., 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar [17] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar [18] I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in System Modeling and Optimization (25th IFIP TC7 Conference, (2013), 328. doi: 10.1007/978-3-642-36062-6_33. Google Scholar [19] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, Second edition, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar [20] G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Ann., 331 (2005), 41. doi: 10.1007/s00208-004-0573-7. Google Scholar [21] M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. Google Scholar [22] M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Appl. Anal., 88 (2009), 1053. doi: 10.1080/00036810903114841. Google Scholar [23] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988). Google Scholar [24] N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1997), 459. Google Scholar [25] J. E. Lagnese, Modelling and stabilization of nonlinear plates,, in Estimation and Control of Distributed Parameter Systems (Vorau, (1990), 247. Google Scholar [26] J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1988). Google Scholar [27] J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl. (9), 85 (2006), 269. doi: 10.1016/j.matpur.2005.08.001. Google Scholar [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [29] T. J. Pedley, The Fluid Mechanics of Large Blood Vessels,, Cambridge University Press, (1980). doi: 10.1017/CBO9780511896996. Google Scholar [30] K. Petersen, Ergodic Theory,, Cambridge Studies in Advanced Mathematics, (1983). doi: 10.1017/CBO9780511608728. Google Scholar [31] B. Schmalfuß, The random attractor of the stochastic Lorenz system,, Z. Angew. Math. Phys., 48 (1997), 951. doi: 10.1007/s000330050074. Google Scholar [32] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-1-4757-5037-9. Google Scholar [33] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, [2013 reprint of the 2001 original] [MR1928881], (2001). Google Scholar [34] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984). Google Scholar [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995). Google Scholar [36] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar
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